After that, upgrade your Power Supply and then buy yourself a nice low endish graphics card, I'd recommend something like a Radeon HD 5570 or 6670.. these cards shouldn't be too expensive and play games really well at lower resolutions (1280x1024 or 1366x768)

^^ Don't listen to this guy, you can get a load of performance out of your PC. see my girlfriends rig for ideas, her PC can play games even like GTA IV, Battlefield 3, etc.

Yes having a SSD is really nice, but I would never recommend that for your needs. Your computer may not be the fastest, but it would definitively benefit from a new GPU (not a top of the line one). With your no name PSU it is hard to recommend anything because its like a time bomb where we don't know when it will fail dragging with it other parts (most likely). As others have suggested, getting a new PSU (a quality 400 or 450w would be plenty) would be a priority, then getting a new GPU would be more appropriate.

Now, if you want to take the risk, I would recommend at least getting a GPU which doesn't require much power. It won't be amazing, but still much better than what you currently have. I am not fully aware of recent models that fit these requirements. I know I did this for my gf for when I want to play games on her computer, I can play fine with lowered details.